There are two closely related constructions, due Yuri Manin, for finitely generated quadratic algebras, and due Tambara, for finite dimensional algebras.

Tambara’s universal coacting bialgebra

If $A$ is a finite dimensional (associative unital) $k$-algebra, and $D$ the functor $D\mapsto D\otimes A$ where $D$ is a $k$-algebras has a left adjoint $a(A,-)$.

where $A,D$ are arbitrary $k$-algebras. $a(A,A)$ has a canonical structure of a coalgebra, making it into a $k$-bialgebra, the universal coacting bialgebra.

Daisuke Tambara, The coendomorphism bialgebra of an algebra, J. Fac. Sci. Univ. Tokyo Sect. IA Math, 37, 425-456, 1990 pdf

Tambara’s construction is dual to the universal measuring coalgebra of Sweedler.

Manin’s universal coacting bialgebra

In a similar way to above, one utilizes the adjunction between inner hom and $!$ functor for quadratic algebras.

Yu. I. Manin, Quantum groups and non-commutative geometry, CRM, Montreal 1988.